The deep decarbonisation of the power sector coupled with electrification of end-use sectors will be crucial towards a carbon neutral economy, as required to achieve the Paris Agreement's goal. Several studies have highlighted the relevance of electrification under deep decarbonisation. However, previous work does not explore what would be the major shifts towards electrification, i.e., in what economic activities it will likely occur and when up to 2050 considering gradually stricter GHG emissions constraints. This is of upmost relevance since relatively small variations in emission caps may trigger substantial modifications in specific components of the energy system, namely the shift for the electrification of a particular energy end-use, with impacts on the power sector's portfolio. In this paper, we analyse the extension of the electrification of the energy system as a cost-effective strategy for deep decarbonisation. We set a large number of increasingly stringent mitigation caps to assess: (i) the degree of electrification of different energy end-uses across all economic activities, (ii) the impact in power sector portfolio and costs and (iii) investment needs. The novelty of this paper relies on the anticipation of electrification of activities traditionally supplied by non-electricity energy carriers, by exploring when and how such transformation may occur in the future, and how much it would cost. We assess the case of Portugal till 2050 by using the TIMES_PT model to generate 50 increasingly stricter decarbonisation scenarios. In the long term, incremental changes (+1%) in more aggressive decarbonisation targets (beyond -70% reduction) induce substantial increase in the share of electrification growth rates. Electric private vehicles, electricity-based steam and heat production in ceramic industrial sector and heat pumps in buildings are the most cost-effective electric technologies. We found that a decarbonisation up to near -80% of 1990's levels of the Portuguese energy system does not have a significant impact on the power sector unit costs, and does not surpass historic values for some years. However, it should be noted that incremental changes (+1%) in more aggressive decarbonisation targets may increase sharply electricity costs in 2050 (+ 9%). Thus, focusing in only few scenarios may narrow the role of electrification (or other mitigation options) and its associated costs for deep decarbonisation. This paper allows researchers, planners and decision makers to enhance awareness regarding the relevance and cost-effectiveness of electrification under decarbonisation, namely its feasibility and affordability, providing fruitful insights.